If you're in the rare situation of using GoDaddy DNS but don't use them as a registrar, then you're in luck. Simply sign up with a new DNS provider. They will give you their DNS servers which you need to set as the DNS servers that are authoritative for your domain. Then sign into your registrar and change the authoritative DNS servers for your domain. There will be a propagation delay but once it's done you're all set.
If you are in the extremely common situation of having registered your domain through GoDaddy and also use their DNS service, then you have a problem because to move to another DNS provider you need to sign into GoDaddy.com to make the change I've described above i.e. change which DNS provider is authoritative for your domain. You can't do this until GoDaddy.com is back online. So what I suggest is that you sign up for a new DNS provider and then keep checking GoDaddy.com. As soon as it comes back online, sign in and make the change to your new provider as quick as you can.
Other data:
Whois requests for godaddy domains are currently failing because whois.godaddy.com is offline due to name resolution failure.
Godaddy's twitter feed is a good source of updates, although they are claiming to be making progress and all my godaddy DNS hosted domains are still offline, so it seems to be more marketing speak than real data: https://twitter.com/godaddy
As mentioned, Anonymous seems to be behind it as three tweets on their twitter account seem to indicate: https://twitter.com/AnonOpsLegion
I don't think the scale of this attack is fully understood yet. According to the CBC, GoDaddy hosts over 5 million websites (not sure if that's DNS, registrar, etc) so expect this to be big news and potentially the next political football.
Edit: And finally, http://www.downforeveryoneorjustme.com/ is down for everyone because it's over quota. Via Reddit which is also covering this: http://www.reddit.com/r/technology/comments/znvwk/godaddycom...